Power Products in Elliptic Divisibility Sequences and Prime-Incidence Obstructions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917546508156928 |
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| author | Kym, Dongyeon |
| author_facet | Kym, Dongyeon |
| contents | Let $E/\mathbb Q$ be an elliptic curve, let $P\in E(\mathbb Q)$ be non-torsion, and let $(D_n)$ be the associated elliptic divisibility sequence. For a fixed prime $ρ$, we study when an arbitrary finite product
\[
\prod_{i=1}^k D_{n_i}
\]
can be a $ρ$-th power in $\mathbb Q^\times$. The main result is that, under the hypothesis that $D_1$ is divisible by $2$ or $3$, such product relations impose rigid restrictions on the large prime divisors of the indices $n_i$. More precisely, for every $B\ge 2$, all sufficiently large prime divisors $\ell$ which occur as simple largest prime divisors of the indices and whose complementary cofactors are $B$-smooth must occur in $ρ$-balanced blocks. Equivalently, the corresponding prime-incidence rows over $\mathbb F_ρ$ have pairwise disjoint supports, are linearly independent, and satisfy the packing bound
\[
|Λ^*|\le \lfloor k/ρ\rfloor .
\]
In particular, if $n_i=\ell_i a_i$, where the $\ell_i$ are sufficiently large primes and the $a_i$ are $B$-smooth, then a $ρ$-th power product relation can hold only if each prime $\ell$ occurs among the $\ell_i$ with multiplicity divisible by $ρ$.
The proof combines Silverman's valuation law, a fixed finite-prime-set consequence of Reynolds' finiteness theorem, and the Hasse bound. The case $ρ=2$ gives the corresponding square-product obstruction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25797 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Power Products in Elliptic Divisibility Sequences and Prime-Incidence Obstructions Kym, Dongyeon Number Theory 11G05 (Primary) 11B37, 11D45, 11D61 (Secondary) Let $E/\mathbb Q$ be an elliptic curve, let $P\in E(\mathbb Q)$ be non-torsion, and let $(D_n)$ be the associated elliptic divisibility sequence. For a fixed prime $ρ$, we study when an arbitrary finite product \[ \prod_{i=1}^k D_{n_i} \] can be a $ρ$-th power in $\mathbb Q^\times$. The main result is that, under the hypothesis that $D_1$ is divisible by $2$ or $3$, such product relations impose rigid restrictions on the large prime divisors of the indices $n_i$. More precisely, for every $B\ge 2$, all sufficiently large prime divisors $\ell$ which occur as simple largest prime divisors of the indices and whose complementary cofactors are $B$-smooth must occur in $ρ$-balanced blocks. Equivalently, the corresponding prime-incidence rows over $\mathbb F_ρ$ have pairwise disjoint supports, are linearly independent, and satisfy the packing bound \[ |Λ^*|\le \lfloor k/ρ\rfloor . \] In particular, if $n_i=\ell_i a_i$, where the $\ell_i$ are sufficiently large primes and the $a_i$ are $B$-smooth, then a $ρ$-th power product relation can hold only if each prime $\ell$ occurs among the $\ell_i$ with multiplicity divisible by $ρ$. The proof combines Silverman's valuation law, a fixed finite-prime-set consequence of Reynolds' finiteness theorem, and the Hasse bound. The case $ρ=2$ gives the corresponding square-product obstruction. |
| title | Power Products in Elliptic Divisibility Sequences and Prime-Incidence Obstructions |
| topic | Number Theory 11G05 (Primary) 11B37, 11D45, 11D61 (Secondary) |
| url | https://arxiv.org/abs/2605.25797 |